Professor Silver:
Of course, the above "proof" is faulty. But, in establishing
that there's at least one sentence, G ("This sentence is unprovable"),
that is true but unprovable, this same model is alluded to. The
model is singled out in order to establish what it is that the
sentence G is true of. I am imagining that Kanovei objects
to this reference to the "standard model" (as being similar
to referring to unicorns), yet this reference is needed to
establish the truth of G.
Matt Insall:
If I understand your question correctly, you would like to eliminate
reference to
the standard model in Gödel's proof of the first incompleteness
theorem. ...As I understand it, the non-constructive version just
produces an instance
of a modified
version of the Liars' Paradox. The proof that I think satisfies your
query
shows the following:
(S) If PA is satisfiable, then PA does not deduce Con(PA).
(S) is the second theorem. So far, Kanovei's reservations have
concerned only the first theorem, though I can easily imagine
additional reservations he might have about the second theorem.
For example, 'Con(PA)' purportedly means "PA is consistent". But,
take away the standard model and 'Con(PA)' doesn't "mean" anything.
It's just complicated gobbledy-gook. Anyway, Kanovei hasn't said
this, so I don't want to put words into his mouth.
As far as I can tell, Kanovei's concerns are not something that
could be eliminated by a mere rephrasing of the proof of Gödel's first
theorem. He believes the standard model is just fiction and won't
accept any notion of truth based on it. It looks to me as though
there are two possibilities: (1) ignore Kanovei's objections, or (2)
take them seriously.
As for (1), Joe Shipman has emphasized that the standard model is
well-defined, and Randall Holmes wrote:
Contrary to what seems obvious to certain other correspondents on
the
list, I think that any child can understand what a standard model
of
PA is like, and it takes a great deal of logical sophistication to
be
talked out of this understanding.
On the other hand, if (2) is thought worthy of attention, it
seems to me the only recourse would be to eliminate any reference to
the standard model when invoking the notion of truth for first-order
sentences of PA. This would require a revamping of the usual truth
definition. As a start, let's say that closed formulas of the
language of PA are flat-out true or flat-out false, depending on
whether they're true of the natural numbers or false of them. So,
(the universal closure of) 'S(x) = S(y) -> x = y' would be flat-out
true. So far, no reference to "models". But, if we wanted to spell
this out thoroughly, how would we continue? What would the metatheory
look like (Would there even be a metatheory?)? Does Kanovei, or
anyone else, have an idea how one might fill in the details?
Charlie Silver